Cubic Surfaces Jaap Top - Wiskunde

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Heinrich Heine Universität, Düsseldorf (1999, Claudia Weber & Ulrich Forster, ceramics) 20
Special property of Clebsch’s diagonal surface S: each of the planes x j + x k = 0 intersects S in a union of three concurrent lines. Point of intersection of three such lines is called Eckardt point. The Clebsch surface is the unique cubic with 10 such points. The Fermat cubic x 3 + y 3 + z 3 + w 3 = 0 is the unique one with 18 such points. All other cubic surfaces have 0, 1, 2, 3, 4, 6 or 9 such points, as Beniamino Segre proved in 1942. F.E. Eckardt: highschool teacher in Chemnitz. He has exactly one paper, published after he died: Mathematische Annalen Vol. 10 (1876) In Utrecht, Tu Nguyen finished his PhD thesis in 2000 concerning Eckardt points. 21
Page 1 and 2: Cubic Surfaces Jaap Top IWI-RuG & D
Page 3 and 4: Problem: fix m ≥ 1, put n := 2 m
Page 5 and 6: S defines a (smooth) cubic surface.
Page 7 and 8: Alfred Clebsch (1833-1872) 7
Page 9 and 10: Clebsch’s diagonal surface 9
Page 11 and 12: Same journal, 1872, August 3: A rep
Page 13: Reproductions of plaster and string
Page 16: July 1890, American Journal of Math
Page 19: Duco van Straten and Stephan Endra
Page 23 and 24: The starting point 1849, Arthur Cay
Page 25 and 26: example: Clebsch diagonal surface x
Page 27 and 28: Theorem: this is possible over R if
Page 29 and 30: Take two conics f = 0 and g = 0 int
Page 31 and 32: N.B.: in the case of a Galois-stabl
Page 33 and 34: So it is possible to improve Euler
Page 35 and 36: Simple practical Maple algorithm fo
Page 37 and 38: Explicit construction of smooth cub
Page 39 and 40: Variant of Manin’s unirationality

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